Practice section 5 of 6

Relative to O, A,B,C have position vectors $\mathbf i+6\mathbf j-2\mathbf k$, $-3\mathbf i+2\mathbf j-5\mathbf k$, $3\mathbf i-2\mathbf j+2\mathbf k$. L passes through A,C.

(i) Find a vector equation of L. [2]

(ii) Find the exact length of projection of AB onto L. [3]

(iii) Hence or otherwise find the shortest distance from B to L in exact form. [2]

D is such that ABCD is a parallelogram. Use a vector product to find its exact area. [3]

Relative to O, A,B,C have position vectors $5\mathbf i+4\mathbf j+10\mathbf k$, $-4\mathbf i+4\mathbf j-2\mathbf k$, $-5\mathbf i+9\mathbf j+5\mathbf k$.

(i) Find the Cartesian equation of AB. [3]

(ii) Find the length of projection of AC onto AB. [3]

(iii) Hence or otherwise find the perpendicular distance from C to AB and the position vector of its perpendicular foot N. [6]

(iv) D lies on CN produced such that N is the midpoint of CD. Find the position vector of D. [2]

Relative to O, A,B,C have position vectors $-3\mathbf i+8\mathbf j+\mathbf k$, $7\mathbf i+3\mathbf j+6\mathbf k$, and $\alpha\mathbf i+\beta\mathbf j-6\mathbf k$ respectively, where α,β are real.

(i) Given $BC:\mathbf r=(7+3\lambda)\mathbf i+3\mathbf j+(6+4\lambda)\mathbf k$, find α,β. [2]

(ii) Find the length of projection of AB onto BC. [2]

(iii) Hence or otherwise find the position vector of the perpendicular foot from A to BC. [3]

(iv) Find the exact area of triangle ABC. [3]

Relative to O, A,B have position vectors $4\mathbf i+3\mathbf j$ and $5\mathbf i+7\mathbf j$. Line $l_1$ passes through B parallel to $\mathbf i+2\mathbf j+\alpha\mathbf k$. Line $l_2$ has Cartesian equation $(1-x)/2=(z-2)/1$, y=4.

(i) Given $l_1,l_2$ are perpendicular, show α=2. [2]

(ii) Find the length of projection of BA onto $l_1$. Hence find vector BN, where N is the perpendicular foot from A onto $l_1$. [4]

(iii) A′ is the reflection of A in $l_1$, and C makes AA′CB a parallelogram. Find its exact area. [3]

Line $l_1$ passes through A=(−1,−2,1), B=(0,1,5). Line $l_2$ has equation $x-1=(y-2)/2=z-3$. They intersect at A. Find:

(i) vector equations of $l_1,l_2$; [2]

(ii) their acute angle; [2]

(iii) the position vector of the perpendicular foot F from B onto $l_2$; [3]

(iv) the equation of the line $l_3$, the mirror image of $l_1$ in $l_2$. [3]

With respect to O, A,B,C have position vectors $\mathbf a=(1,\sin t,0)$, $\mathbf b=(\cos t,2,1)$, $\mathbf c=(2,-1,1)$, where $0\le t<\pi$.

(a) Find the exact t given a is perpendicular to b. [3]

(b) X is on AB produced with AB:BX=1:4, and ACXY is a parallelogram. Take $t=\pi/2$ for the rest of the question.

(i) Find the position vectors of X,Y. [4]

(ii) Find the area of ACXY. Hence find the shortest distance from X to line AC. [3]

(iii) Line $l_1$ has equation $(x-2)/6=(3-y)/h$, z=k, where h,k are constants. The line through A,B intersects $l_1$ at a right angle. Find h,k. [4]